In each of the figures 1A and 1B below, construct the perpendicular bisector of AB in the appropriate model. As usual, write a FEW main steps to make the construction clear. Do not write a lot.
In this D-model figure, construct the D-perpendicular bisector of AB. (The point I is the one removed in the model.)
In this P-model figure, construct the P-perpendicular bisector of AB.
· Review of terminology: The circles of Apollonius of points A and B are the same as the hyperbolic pencil (hyperbolic coaxal family) with limit points A and B. These are the circles orthogonal to the elliptic pencil (elliptic coaxal family) of circles through A and B. The other kind of pencil or coaxal family is a parabolic pencil tangent to a given line at a given point A.
· The task: In each case in the table below, the family of objects is either a set of circles that form a pencil or else are arcs of such circles. For each case below, you are asked to tell what is the pencil? This means to tell (1) which of the 3 types of pencil and (2) tell specifically what are the special points A and B for the pencil). You may wish to sketch figures to help in your explanation.
· Notation: In these questions, the disk for the P-model is the interior of the circle c with center O and radius r. The DWEG model consists of points in the plane with the point I removed (and the point Infinity added).
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Description of the family |
What kind of pencil? |
What are the 2 points A and B? (One point and line for parabolic) |
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(a) In the P-model, for a P-point G, what pencil is the set of P-lines through G? |
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(b) In the D-model, for a D-point H, what pencil is the set of D-lines through H? |
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(c) In the P-model, for P-points U and V, what pencil is the set of P-circles with P-center U? |
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(d) In the D-model, for D-points X and Y, what pencil is the set of D-circles with D-center X? |
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(e) In the P-model, for a P-line g, what pencil is the set of P-lines orthogonal to g? |
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(f) In the D-model, for a D-line h, what pencil is the set of D-lines orthogonal to h? |
Consider a figure with two lines AB and CD, with A and D on opposite sides of line BC.
Is this statement True or False?
Statement: For this figure in the hyperbolic plane, if angle ABC = angle DBA, then line AB is parallel to line CD.
· True or False? _________________
· If true, indicate why (the key idea without all the details). If false, indicate why or give a counterexample
· State the converse of the statement. Is the converse true or false in the hyperbolic plane?